The major purpose of this research is to develop new methods for the analysis of time to event data that are encountered in cancer research. Specifically, we propose the use of the accelerated failure time model for relating the effect of time dependent covariates to survival. We will demonstrate how these models could be used for analyzing the effect of treatment in a randomized clinical trial when compliance problems are an issue. Semiparametric estimates of the parameters will be obtained by using rank methods for censored data. The theoretical properties of these estimates will be established using counting processes and martingale methods. Computer algorithms will be developed for finding these estimates in an efficient manner and the small sample properties will be assessed by computer simulation. We will also develop tests for comparing multiple time to event data that are subject to right censoring among different treatments. Finally, we shall use survival methods to study the problem of constructing group sequential tests to compare response rates when there is lag time in reporting.